Number of real roots and points of intersection.
On the number of real roots of random polynomials.
"In this talk I´ll address the problem of counting roorts of random polynomials, which is an important research area of Mathematics and in some areas of Physics. For almost a century a considerable amount of literature about this problem has emerged from fields as probability, geometry, algebraic geometry, algorithm complexity, quantum physics, etc. In spite of its rich history it is still an extremely active field. In this talk we will review some old and new results on this topic, in particular, the breakthrough result by Shub and Smale".
On the number of roots of sparse polynomial systems.
"How many common roots does one expect a system of 'n' polynomials in 'n' variables to have? I will try to convince you that the Bézout bound (product of the degrees) is usually the wrong answer. Then I will review sharper root counting theorems, mostly for sparse complex polynomial systems.
The expected number of real roots for sparse real polynomial systems is mostly uncharted territory".
Partial Hyperbolicity and Ergodic Theory.
Some aspects of partially hyperbolic diffeomorphisms in dimension 3.
"I will present some general results that concern the geometry and dynamics of partially hyperbolic diffeomorphisms in 3D manifolds. On some classes of such diffeomorphisms, one can obtain unconditional results on ergodicity or rigidity of uu-states".
On the growth rate inequality for sphere endomorphisms.
"The following assertion was conjectured by M. Shub: Let f be a C1 map of the two-sphere having degree d>1 and p_n denote the number of fixed points of f^n. Then, p_n increases exponentially with n with rate log d. I will make an account of some examples, techniques and results used in some particular cases."
Dynamical Systems and Cellular Biology.